ELECTRON-CAPTURE AND β-DECAY RATES FOR sd-SHELL NUCLEI IN STELLAR ENVIRONMENTS RELEVANT TO HIGH-DENSITY O–NE–MG CORES

The evolution and final fates of stars depend on their initial masses M_I (e.g., Nomoto et al. 2013), which are still subject to some uncertainties involved in stellar mass loss, mixing processes, and nuclear transition rates. Stars with M_I = 0.5–8 M_⊙ form a C–O core after He burning and become C+O white dwarfs. Stars with M_I > 10 M_⊙, on the other hand, form an Fe core and later explode as core-collapse supernovae. In stars with M_I = 8–10 M_⊙, an O–Ne–Mg core is formed after carbon burning. Stars with M_I = 8–10 M_⊙ can end up in various ways, that is, as O–Ne–Mg white dwarfs without explosions, or as electron-capture supernovae or Fe core-collapse supernovae (Miyaji et al. 1980; Nomoto 1984, 1987; Nomoto & Hashimoto 1988).

The evolutionary changes in the central density and temperature of the O–Ne–Mg core are determined by the competition among the contraction, cooling, and heating processes. If heating is fast enough relative to contraction, the temperature would become high enough to ignite Ne–O burning, which would result in the Fe core formation. If cooling is fast, the contraction would lead to the collapse of the O–Ne–Mg core and an electron-capture supernova. The fate of the stars with M_I ∼ 8–10 M_⊙ is thus determined by these competitions. Since these competing processes are induced by electron capture and β-decay, the accurate weak rates are crucially important.

We study electron-capture and β-decay rates in sd-shell nuclei at stellar environments relevant to the final evolution of electron-degenerate O–Ne–Mg cores in stars with the initial masses of M_I = 8–10 M_⊙. Nuclear Urca processes, especially in nuclear pairs with A = 23 and 25, are important for the cooling of the O–Ne–Mg cores after carbon burning (Jones et al. 2013; Toki et al. 2013). Electron-capture reactions and successive gamma emissions in nuclei with A = 24 and 20 are important for the contraction and heating of the core in later stages, leading to an electron-capture supernova. The cooling and heating rates of the core, as well as the core contraction rate, are determined by the weak rates for the nuclear pairs. The fate of the stars depends sensitively on the nuclear electron-capture and β-decay rates.

Precise evaluations of the weak rates in stars are quite important for the evolutions of not only the stars having O–Ne–Mg cores but also lower-mass stars having C–O cores (which contain primordial species of A = 20–27). The rates are also important for the final fates of O–Ne–Mg, C–O, and hybrid C–O–Ne white dwarfs in binary systems. We thus provide electron-capture and β-decay rates at stellar environments with such fine meshes as Δlog₁₀(ρY_e) = 0.02 (Y_e) is the electron mole number and ρ is the nucleon density in units of g cm⁻³ and Δlog₁₀ T = 0.05 (T is the temperature in units of K) at various densities and temperatures, so that they can be used for the studies of astrophysical processes sensitive to the rates, as well as the determination of the final fates of stars. We provide the table for the weak rates on the web (Suzuki et al. 2015).

In Section 2, Q-values in sd-shell nuclei and relations to triggering electron-capture reactions are discussed. In Section 3, the formalism for electron-capture and β-decay rates and the neutrino-energy-loss rate and γ-ray heating rate are given. Coulomb corrections are also taken into account. In Section 4, the weak rates and energy loss by neutrino emissions for nuclear pairs with A = 23, 25, and 27, which are important for the Urca processes, are studied. A summary is given in Section 5.

The triggering of electron-capture reactions is determined by Q-values of nuclear transitions and values of the chemical potential of electrons at high densities and high temperatures. The Q-values for β-decays in sd-shell nuclei with A = 17–31 are shown in Figure 1. They are taken from Wang et al. (2012).

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In general, the Q-values can become small for odd-A nulei compared to even-A nuclei, as can be noticed from Figure 1. They are particularly as small as 1.5–4.4 MeV for the pairs with A = 23, 25, 27, 29, and 31: ²³Ne $\to$ ²³Na, ²⁵Na $\to$ ²⁵Mg, ²⁷Mg $\to$ ²⁷Al, ²⁹Al $\to$ ²⁹Si, and ³¹Si $\to$ ³¹P. The Urca processes can occur for these nuclear pairs if the transition from the ground state (GS) of the mother nucleus to the GS of the daughter nucleus is not forbidden. In fact, the Urca processes for the pairs ²³Ne $\leftrightarrow$ ²³Na and ²⁵Na $\leftrightarrow$ ²⁵Mg are found to be important for the cooling of the O–Ne–Mg core of stars with 8–10 M_⊙ (Jones et al. 2013; Toki et al. 2013). As the GS-to-GS transition for ²⁷Mg $\leftrightarrow$ ²⁷Al is a forbidden transition, the Urca process does not occur for this pair (Toki et al. 2013), though the Q-value is very small: Q = 2.61 MeV. The order of the occurrence of the Urca processes depends on the Q-value of the transition. The Urca process for the pair ²⁵Na–²⁵Mg with Q = 3.83 MeV takes place first, and it is followed by the Urca process for the pair ²³Ne–²³Na with Q = 4.38 MeV.

For even-A nuclei, the β-decay Q-values are generally large for transitions from odd–odd nuclei to even–even nuclei, while they are relatively small for transitions from even–even to odd–odd nuclei, owing to the pairing effects. In later stages of the evolution of O–Ne–Mg core stars, electron-capture reactions on even–even nuclei such as ²⁴Mg with Q-value of 5.52 MeV and ²⁰Ne with Q-value of 7.02 MeV are triggered at higher densities. They are succeeded by electron-capture reactions on ²⁴Na and ²⁰F, respectively. These odd–odd nuclei have small threshold energies for the electron-capture reactions, 2.47 MeV for ²⁴Na and 3.81 MeV for ²⁰F. The double electron-capture reactions on ²⁴Mg and ²⁰Ne become important for heating stars in later stages of star evolutions.

Here we consider cooling of the O–Ne–Mg core by Urca processes in nuclear pairs with A = 23, 25, and 27 and heating of stars by electron-capture reactions on nuclei with A = 20 and 24. The cooling process occurs at log₁₀(ρY_e) = 8.8–9.0 first for ²⁵Na–²⁵Mg and ²³Ne–²³Na pairs, and then the secondary one occurs at log₁₀(ρY_e) = 9.3–9.8 for other pairs with the second-smallest Q-values in each mass number A = 23, 25, and 27 (Jones 2013). The heating process occurs at log₁₀(ρY_e) = 9.2–9.7 for captures on ²⁴Mg and ²⁰Ne. The successive electron captures on ²⁴Na and ²⁰F with smaller Q-values occur at log₁₀(ρY_e) = 8.2 and 8.8. In the present work, the electron-capture and β-decay rates in these sd-shell nuclei that concern the cooling of the O–Ne–Mg core after C burning and the heating before O burning are evaluated covering the density region mentioned above, log₁₀(ρY_e) = 8.0–10.0, for relevant temperatures. The weak rates for nuclei with A = 17–28, as well as neutrino-energy-loss rates and γ-ray heating rates, are obtained for densities log₁₀(ρY_e) = 8.0–11.0 in fine steps of 0.02 and temperatures log₁₀ T = 8.0–9.65 and 7.0–8.0 in steps of 0.05 and 0.20, respectively (Suzuki et al. 2015).

The electron-capture and β-decay transitions are dominantly induced by Gamow–Teller (GT) transitions in the present stellar conditions with densities log₁₀(ρY_e) = 8 – 10 and temperatures log₁₀T = 8 – 9.6. Structures of sd-shell nuclei are well described by a shell-model Hamiltonian, USD (Brown & Wildenthal 1983, 1987, 1988). Energy spectra, magnetic dipole (M1), and GT strengths with quenched spin g-factor and axial-vector coupling constant, g_A, as well as Q moments and E2 transition strengths with effective charges, are systematically well reproduced with USD. The Hamiltonian has been updated by taking into account recent data. While neutron-rich O and F isotopes were too tightly bound for USD, the new versions, USDA and USDB (Brown & Richter 2006; Richter et al. 2008), are free from this problem. Here USDB is used for the evaluation of $B(\mathrm{GT})$ with ${g}_{{\rm{A}}}^{{\rm{eff}}}/$g_A = 0.764.

Electron-capture rates at high densities and high temperatures are evaluated as (Fuller et al. 1980, 1982a, 1982b, 1985; Langanke & Martinez-Pinedo 2001, 2003; Suzuki et al. 2011)

$$\begin{eqnarray}\lambda & = & \displaystyle \frac{\mathrm{ln}2}{6146(s)}\displaystyle \sum _{i}{W}_{i}\displaystyle \sum _{f}({B}_{{if}}(\mathrm{GT})+{B}_{{if}}(F)){{\rm{\Phi }}}^{\mathrm{ec}}({Q}_{{if}})\\ {{\rm{\Phi }}}^{\mathrm{ec}}({Q}_{{if}}) & = & {\displaystyle \int }_{{\omega }_{\mathrm{min}}}^{\infty }\omega p{({Q}_{{if}}+\omega )}^{2}F(Z,\omega ){S}_{{\rm{e}}}(\omega )d\omega ,\\ {Q}_{{if}} & = & ({M}_{{\rm{p}}}{c}^{2}-{M}_{{\rm{d}}}{c}^{2}+{E}_{i}-{E}_{f})/{m}_{{\rm{e}}}{c}^{2},\\ {W}_{i} & = & (2{J}_{i}+1){e}^{-{E}_{i}/{kT}}/\displaystyle \sum _{i}(2{J}_{i}+1){e}^{-{E}_{i}/{kT}},\end{eqnarray} \tag{ 1 }$$

where ω and p are electron energy and momentum in units of m_ec² and m_ec, respectively, M_p and M_d are nuclear masses of parent and daughter nuclei, respectively, and E_i and E_f are excitation energies of initial and final states, respectively. Here B(GT) and B(F) are the GT and Fermi strengths, respectively, given by

$$\begin{eqnarray}{B}_{{if}}(\mathrm{GT}) & = & {({g}_{A}/{g}_{V})}^{2}\displaystyle \frac{1}{2{J}_{i}+1}{| \langle f| | \displaystyle \sum _{k}{\sigma }^{k}{t}_{+}^{k}| | i\rangle | }^{2}\\ {B}_{{if}}(F) & = & \displaystyle \frac{1}{2{J}_{i}+1}{| \langle f| | \displaystyle \sum _{k}{t}_{+}^{k}| | i\rangle | }^{2},\end{eqnarray} \tag{ 2 }$$

where J_i is the total spin of initial state and ${t}_{+}| p\gt =| n\gt .$ In the case of β-decay, t₊ is replaced by t₋: ${t}_{-}| n\gt =| p\gt .\;$F(Z, ω) is the Fermi function, and S_e(ω) is the Fermi–Dirac distribution for electrons, where the chemical potential, μ_e, is determined from the density, ρY_e, by

$$\begin{eqnarray}&&\rho {Y}_{{\rm{e}}}=\displaystyle \frac{1}{{\pi }^{2}{N}_{{\rm{A}}}}{(\displaystyle \frac{{m}_{{\rm{e}}}c}{{\rm{\hslash }}})}^{3}{\displaystyle \int }_{0}^{\infty }({S}_{{\rm{e}}}-{S}_{{\rm{p}}}){p}^{2}{dp}\end{eqnarray} \tag{ 3 }$$

where N_A is the Avogadro number and S_p is the Fermi–Dirac distribution for positrons with the chemical potential μ_p = −μ_e. It can become as large as 2–5 (11) MeV at high densities log₁₀ ρY_e = 8 – 9 (10). It slightly decreases as the temperature increases. The reaction rates become larger at higher densities because of the large chemical potential.

Since it is sometimes important to include GT transitions from thermally populated excited states in the steller interior, we include transitions from the excited states of the parent nucleus in addition to the GS with the partition function W_i. When temperature becomes high and the excitation energies of excited states of the parent nucleus are low, the nucleus can be thermally excited and the population of the excited states can be large. In such a case, the transitions from the excited states can give important contributions to the capture rates. In the present work, excited states with excitation energies up to E_x = 2 MeV are taken into account.

In the case of β-decays, the integral in Equation (1) is replaced by

$$\begin{eqnarray}&&{{\rm{\Phi }}}^{\beta }({Q}_{{if}})={\displaystyle \int }_{1}^{{Q}_{{if}}}\omega p{({Q}_{{if}}-\omega )}^{2}F(Z+1,\omega )(1-{S}_{{\rm{e}}}(\omega ))d\omega .\end{eqnarray} \tag{ 4 }$$

The β-decay rates decrease as the density increases owing to the blocking of the electron density in contrast to the electron-capture case. There is, therefore, a density where both the electron-capture and β-decay rates are balanced. When such densities depend little on tempertures, both the transitions occur simultaneously, emitting both ν and $\bar{\nu }$ at such densities, which we call "Urca densities." Here the rates are evaluated for 8.0 < log₁₀(ρY_e) < 11.0 in fine steps of 0.02 and 8.0 < log₁₀ T < 9.65 in steps of 0.05 and also for 7.0 < log₁₀ T < 8.0 in steps of 0.20.

We give also formulae for the energy loss by neutrino emissions and heating by γ emissions for astrophysical applications. The neutrino-energy-loss rate is given by

$$\begin{eqnarray}\xi & = & \displaystyle \frac{(\mathrm{ln}2){m}_{{\rm{e}}}{c}^{2}}{6146(s)}\displaystyle \sum _{i}{W}_{i}\displaystyle \sum _{f}({B}_{{if}}(\mathrm{GT})+{B}_{{if}}(F))\\ & & \times {{\rm{\Psi }}}^{\mathrm{ec}}({Q}_{{if}})\\ {{\rm{\Psi }}}^{\mathrm{ec}}({Q}_{{if}}) & = & {\displaystyle \int }_{{\omega }_{\mathrm{min}}}^{\infty }\omega p{({Q}_{{if}}+\omega )}^{3}F(Z,\omega ){S}_{{\rm{e}}}(\omega )d\omega \end{eqnarray} \tag{ 5 }$$

for electron-capture reactions, while for β-decay transitions Ψ^ec is replaced by

$$\begin{eqnarray}&&{{\rm{\Psi }}}^{\beta }({Q}_{{if}})={\displaystyle \int }_{1}^{{Q}_{{if}}}\omega p{({Q}_{{if}}-\omega )}^{3}F(Z+1,\omega )(1-{S}_{{\rm{e}}}(\omega ))d\omega .\end{eqnarray} \tag{ 6 }$$

The average energy of the emitted neutrino is obtained by the ratio

$$\begin{eqnarray}&&\langle {E}_{\nu }\rangle =\displaystyle \frac{\xi }{\lambda }.\end{eqnarray} \tag{ 7 }$$

The γ-ray heating rate is given by (Takahara et al. 1989; Oda et al. 1994)

$$\begin{eqnarray}\eta & = & \displaystyle \frac{(\mathrm{ln}2){m}_{{\rm{e}}}{c}^{2}}{6146(s)}\displaystyle \sum _{i}{W}_{i}\displaystyle \sum _{f}({B}_{{if}}(\mathrm{GT})+{B}_{{if}}(F)){{\rm{\Gamma }}}^{\mathrm{ec}}({Q}_{{if}})\\ {{\rm{\Gamma }}}^{\mathrm{ec}}({Q}_{{if}}) & = & {\displaystyle \int }_{{\omega }_{\mathrm{min}}}^{\infty }\omega p{({Q}_{{if}}+\omega )}^{2}{E}_{f}{/m}_{{\rm{e}}}{c}^{2}F(Z,\omega ){S}_{{\rm{e}}}(\omega )d\omega \end{eqnarray} \tag{ 8 }$$

for electron-capture reactions, while for β-decay transitions Γ^ec is replaced by

$$\begin{eqnarray}{{\rm{\Gamma }}}^{\beta }({Q}_{{if}}) & = & {\displaystyle \int }_{1}^{{Q}_{{if}}}\omega p{({Q}_{{if}}-\omega )}^{2}{E}_{f}{/m}_{{\rm{e}}}{c}^{2}F\\ & & \times (Z+1,\omega )(1-{S}_{{\rm{e}}}(\omega ))d\omega .\end{eqnarray} \tag{ 9 }$$

The average energy of the emitted γ is obtained by the ratio

$$\begin{eqnarray}&&\langle {E}_{\gamma }\rangle =\displaystyle \frac{\eta }{\lambda }.\end{eqnarray} \tag{ 10 }$$

Here the excited states of the daughter nucleus are assumed to emit γ's, leading to its GS (Takahara et al. 1989; Oda et al. 1994).

Next, the Coulomb corrections on the transition rates due to the electron background are studied. There are two Coulomb effects: one is the screening effects of electrons, and the other is the change of threshold energy due to the change of chemical potential of the ions.

The Coulomb potential is modified owing to the screening effects of relativistically degenerate electron liquid. The modification can be evaluated by using the dielectric function obtained by random phase approximation (Itoh et al. 2002). This effect is included by reducing the chemical potential of electrons by an amount equal to the modification of the Coulomb potential at the origin, ${V}_{s}(0)$ (Juodagalvis et al. 2010), where

$$\begin{eqnarray}{V}_{s}(r) & = & {{Ze}}^{2}(2{k}_{F})J(r)\\ J(r) & = & \displaystyle \frac{1}{2{k}_{F}r}[1-\displaystyle \frac{2}{\pi }\displaystyle \int \displaystyle \frac{\mathrm{sin}(2{k}_{F}{qr})}{{q}^{2}\epsilon (q,0)}{dq}].\end{eqnarray} \tag{ 11 }$$

The screening coefficient J is tabulated in Itoh et al. (2002) (see Toki et al. 2013 for more details). The electron-capture and β-decay rates are slightly reduced and enhanced, respectively, by the effect.

The other effect is caused by the correction of the chemical potential of the nucleus with charge number Z, ${\mu }_{{\rm{C}}}(Z),$ due to the interactions of the nucleus with the electron background (Slattery et al. 1982; Ichimaru 1993). The threshold energy changes by

$$\begin{eqnarray}&&{\rm{\Delta }}{Q}_{{\rm{C}}}={\mu }_{{\rm{C}}}(Z-1)-{\mu }_{{\rm{C}}}(Z),\end{eqnarray} \tag{ 12 }$$

which gets larger for electron-capture processes. The Coulomb chemical potential of a nucleus with Z in a plasma of electron number density n_e and temperature T is given by

$$\begin{eqnarray}&&{\mu }_{{\rm{C}}}(Z)={kTf}({\rm{\Gamma }}),\end{eqnarray} \tag{ 13 }$$

with Γ = Z^5/3Γ_e, Γ_e = $\displaystyle \frac{{e}^{2}}{{{kTa}}_{{\rm{e}}}}$, and a_e = ${(\displaystyle \frac{3}{4\pi {n}_{{\rm{e}}}})}^{1/3}.$ The function f for Γ > 1 is given in Ichimaru (1993). More details are given in Toki et al. (2013). This effect results in the reduction of the electron-capture rates and the enhancement of the β-decay rates.

The contribution of the second effect is larger than the first one, and thus the electron-capture (β-decay) rates are reduced (enhanced). The Urca density is shifted to a larger density region, as shown in the next section.

4.1. ²³Ne–²³Na and ²⁵Na–²⁵Mg Pairs

The electron-capture and β-decay rates for A = 23, 25, and 27 are evaluated at densities and temperatures of stellar environments. The rates for the pair ²⁵Na–²⁵Mg are shown in Figure 2. Transitions from 5/2 ${}_{{\rm{g}}.{\rm{s}}.}^{+},$ 1/2 ${}^{+},$ 3/2 ${}^{+},$ 7/2⁺, and second 5/2⁺ states of ²⁵Mg are included for the electron-capture reactions, while 5/2${}_{{\rm{g}}.{\rm{s}}.}^{+},$ 3/2⁺, and 1/2⁺ states of ²⁵Na are taken into account for the β-decays. Here the Coulomb effects are not included. The electron-capture rates increase as the density and the chemical potential increases. The β-decay rates, on the other hand, decrease as the density increases owing to the blocking of the decays by the electrons with high chemical potential. The Urca density, where both the rates coincide, is found at log₁₀(ρY_e) = 8.77. Dependence of the density on the temperature is quite small: Δlog₁₀(ρY_e) < 0.01 for log₁₀ T = 8–9.2. The Urca density is also found at log₁₀(ρY_e) = 8.92 for the pair with A = 23, ²³Ne–²³Na. The temperature dependence is also as small as Δlog₁₀ ρY_e ≈ 0.01 for log₁₀ T = 8–9.2. It is important to evaluate the rates with fine meshes to obtain the clear Urca densities. Detailed discussion is given in Toki et al. (2013). In the case of the pair with A = 27, ²⁷Mg–²⁷Al, where the GT transition from the initial GS to the final GS does not occur because of a spin difference larger than 1, it is difficult to assign an Urca density as a result of the Q-value mismatch in the two transitions. The densities where electron-capture and β-decay rates coincide depend much on the temperature for the A = 27 case.

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Here we comment on the choice of the steps of the mesh in log₁₀(ρ Ye) and log₁₀ T. Fine steps of 0.02 and 0.05 are adopted for both log₁₀(ρY_e) and log₁₀ T, respectively, as it is not easy to obtain accurate rate values from the interpolation procedure among quantities that differ by many orders of magnitude. For example, from the rates at log₁₀(ρY_e) = 7, 8, 9, 10, and 11 only for a fixed temperature, it is not possible to get an accurate rate by interpolation procedures. The error in the rate can be larger than a factor of 10.

It is true that a procedure using effective log ft values proposed by Fuller et al. (1985) works well for certain cases. The procedure is based on the fact that the change of the rates by orders of magnitude comes mainly from the phase space factor, and the remaining part including the nuclear transition probability does not change drastically. As the phase space factor for the transition between GSs is taken in the procedure, the effective log ft method is valid for the cases where GT transitions between the GSs are dominant. In the case in which the GS-to-GS transitions are forbidden or transitions from excited states give essential contributions, the method becomes invalid when it is applied to the interpolation with a sparse grid of densities at log₁₀(ρY_e) = 7, 8, 9, 10, and 11. The pairs ²⁷Al–²⁷Mg and ²⁰F–²⁰Ne are such examples.

In a density interval of Δlog₁₀(ρY_e) = 0.02 or in a temperature interval of Δlog₁₀ T = 0.05, the rate (log₁₀(rate)) can change by 3–4 orders of magnitude (by 3–4) at most (see Figure 2). For this order of the change in log₁₀(rate) in the relevant intervals, the interpolation can be safely done with an accuracy of Δlog₁₀(rate) = 0.001–0.002. The steps of the meshes chosen here are sufficient for obtaining the rates with further finer meshes by the interpolation procedures.

Next, we compare the present results obtained with USDB with those of Oda et al. (1994). In Oda et al. (1994), USD interaction was used and experimental energies and GT transition rates available were also taken into account (Brown & Wildenthal 1985). Calculated GT strengths, as well as their cumulative sums for USDB and USD, are shown in Figure 3 for the transitions ²⁵Mg $\to$ ²⁵Na and ²³Na $\to$ ²³Ne. Experimental transition strengths to the GSs of the daughter nuclei are also shown. In both ²⁵Mg and ²³Na, the GT strength is more spread for USDB compared with USD and larger strength remains more in the higher excitation energy (E_x) region for USDB. The GT strengths differ little at low excitation energy region E_x < 3 MeV for ²⁵Mg, while for ²³Na their difference in the magnitude is noticed at the low-E_x region below 3 MeV. Calculated B(GT) values for the transition to the GS of ²⁵Na for both USDB and USD are consistent with the experimental value. The latter is only a bit larger than the values of USDB (USD), by 15% (19%). Transition rates are little affected by the use of experimental data for the pair ²⁵Mg–²⁵Na. The present calculated rates with USDB are close to the rates given in Oda et al. (1994).

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In the case of ²³Na, calculated B(GT) values for the transition to the GS of ²³Ne are smaller than the experimental value by factors of 2.67 and 1.96 for USDB and USD, respectively. Use of experimental B(GT) values enhances the transition rates for the pair ²³Na–²³Ne by about a factor of 2.5. When experimental data are taken into account, the rates for USDB become very close to those by Oda et al. (1994), as shown in Table 1. Effects of the use of the experimental B(GT) values are important. Though the rms deviation of calculated GT matrix elements from experimental ones is as small as 0.114 for USDB with the quenching factor (Richter et al. 2008), experimental B(GT) values and experimental excitation energies are taken into account here (Suzuki et al. 2015) when they are available (Endt 1998; Tilley et al. 1998; NNDC 2015).

Table 1.  Transition Rates for the ²³Na–²³Ne Pair

		Electron-capture Rates				β-decay Rates
ρYe	T/109	USDB	USDB*	USD	USD*	USDB	USDB*	USD	USD*
108	0.1	⋯	⋯	⋯	⋯	−2.418	−2.048	−2.284	−2.064
	0.4	−37.243	−36.899	−37.108	−36.913	−2.417	−2.048	−2.284	−2.063
	1.0	−17.640	−17.261	−17.505	−17.278	−2.414	−2.044	−2.280	−2.060
	2.0	−10.644	−10.256	−10.508	−10.270	−2.395	−2.028	−2.257	−2.042
109	0.1	−4.762	−4.333	−4.626	−4.352	−24.294	−23.866	−24.158	−23.185
	0.4	−4.710	−4.282	−4.574	−4.299	−10.700	−10.271	−10.564	−10.290
	1.0	−4.461	−4.046	−4.324	−4.064	−7.265	−6.854	−7.102	−6.863
	2.0	−3.965	−3.580	−3.824	−3.597	−5.275	−5.015	−5.073	−4.983

Note. USDB* denotes that available experimental data are taken into account. Values for USD* are taken from Oda et al. (1994).

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Effects of the Coulomb corrections on the transition rates are studied for the nuclear pairs ²³Ne–²³Na and ²⁵Na–²⁵Mg. The calculated rates for a temperature log₁₀ T = 8.7 are shown in Figure 4 for the pairs with and without the Coulomb effects. The Urca density is shifted toward a higher-density region by Δlog₁₀(ρY_e) = 0.04. Though the change of the Urca density is rather modest, it can affect the fate of the stars with mass around 9 M_⊙.

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Now we discuss energy loss by neutrino emissions and heating by γ emissions during the electron-capture and β-decay processes. Averaged energy of emitted neutrinos defined by Equation (7), $\langle {E}_{\nu }\rangle$, and averaged energy production

$$\begin{eqnarray}&&\langle {E}_{\mathrm{prod}}\rangle ={\mu }_{{\rm{e}}}-{Q}_{\mathrm{nucl}}-\langle {E}_{\nu }\rangle \end{eqnarray} \tag{ 14 }$$

with Q_nucl = M_dc²–M_pc², which is the energy difference between the GSs of daughter and parent nuclei, for electron-capture processes and

$$\begin{eqnarray}&&\langle {E}_{\mathrm{prod}}\rangle -{\mu }_{{\rm{e}}}-\langle {E}_{\nu }\rangle \end{eqnarray} \tag{ 15 }$$

for β-decay processes are shown in Figures 5 and 6 for ²⁵Na–²⁵Mg and ²³Ne–²³Na pairs, respectively. The energy generation in the star is determined by (Martinez-Pinedo et al. 2014)

$$\begin{eqnarray}&&{kT}\displaystyle \frac{{ds}}{{dt}}=\displaystyle \frac{{{dY}}_{{\rm{e}}}}{{dt}}\langle {E}_{\mathrm{prod}}\rangle \end{eqnarray} \tag{ 16 }$$

where s is the entropy per nucleon. The condition that the timescale for thermodynamical equilibrium is shorter than the timescale for the weak interaction is assumed to be satisfied.

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Averaged neutrino energies, $\langle {E}_{\mathrm{prod}}\rangle$, and averaged energy productions, $\langle {E}_{\mathrm{prod}}\rangle$, for electron-capture reactions on ²⁵Mg and ²³Na and β-decay transitions from ²⁵Na and ²³Ne are shown in Figures 5 and 6. The value of $\langle {E}_{\nu }\rangle$ is almost determined by the values of the electron chemical potential and the threshold energy Q_nucl. In the case of electron-capture reactions on ²⁵Mg and ²³Na, the neutrino energy loss increases above the Urca densities and the increase of the averaged energy production starts to be suppressed just at the densities, where the energy production becomes positive. In the case of β-decay transitions from ²⁵Na and ²³Ne, the energy production is suppressed owing to neutrino emissions below the Urca densities, while when it becomes negative at the Urca densities the negative energy production, that is, the energy loss, begins to increase monotonically as the density increases. In both cases of electron-capture and β-decay processes, the energy production is suppressed by neutrino emissions when it becomes positive. In the case with the Coulomb effects, Q_nucl is replaced by Q_nucl +ΔQ_C and μ_e by ${\mu }_{{\rm{e}}}-{V}_{s}(0).$ As ΔQ_C is positive and its magnitude is larger than $| {V}_{s}(0)| ,$ $\langle {E}_{\mathrm{prod}}\rangle$ gets reduced (enhanced) for the electron-capture (β-decay) case. This change of $\langle {E}_{\mathrm{prod}}\rangle$, however, is suppressed above (below) the Urca densities for the electron-capture (β-decay) case since $\langle {E}_{\nu }\rangle$ with the screening effects is smaller (larger) than that without the effects. These features are seen in Figures 5 and 6. We will show calculated results with the Coulomb effects hereafter.

4.2. ²³F–²³Ne, ²⁵Ne–²⁵Na, and ²⁷Na–²⁷Mg Pairs

After the cooling of the O–Ne–Mg core of the stars occurs by nuclear Urca processes for pairs with A = 25 and 23, successive electron-capture reactions are expected to be triggered at higher densities corresponding to the Q-values. Electron-capture reactions on ²⁴Mg with Q = 5.52 MeV are considered to be important in later stages of the star evolutions, leading to electron-capture supernova explosions.

The products of electron-capture and β-decay rates for various nuclear pairs are shown in Figure 7. The peak position for each pair is determined by the Q-value of each transition. The Urca densities for the pairs ²⁴Na–²⁴Mg, ²⁵Ne–²⁵Na, ²³F–²³Ne, and ²⁷Na–²⁷Mg increase in this order. For the pair ²⁰F–²⁰Ne, GT transition cannot occur between the GSs as the GS of ²⁰F is 2⁺ while that of ²⁰Ne is 0⁺. In this case, the contributions from the forbidden transitions between the GSs are taken into account (Martinez-Pinedo et al. 2014). The peak at lower density at log₁₀(ρY_e) ≈ 9.2 corresponds to the transition between ²⁰Ne (2⁺) and ²⁰F (2⁺), while the peak at higher density at log₁₀(ρY_e) ≈ 9.7 corresponds to the transition between ²⁰Ne (0⁺) and ²⁰F (1⁺). The peak at log₁₀(ρY_e) ≈ 9.5 corresponds to the forbiden transition between the GSs. For ²⁴Mg and ²⁰Ne, the heating of stars occurs by double electron-capture reactions due to the pairing effects, that is, by successive electron-capture reactions on ²⁴Na and ²⁰F following the capture reactions on ²⁴Mg and ²⁰Ne, respectively.

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Averaged neutrino energies, $\langle {E}_{\nu }\rangle$, and averaged energy productions, $\langle {E}_{\mathrm{prod}}\rangle$, for electron-capture reactions on ²⁵Na and β-decay transitions from ²⁵Ne are shown in Figure 8. Here transitions from 5/2${}_{{\rm{g}}.{\rm{s}}.}^{+},$ 3/2⁺, and 1/2⁺ states of ²⁵Na are included for the electron-capture reactions, while those from 1/2${}_{{\rm{g}}.{\rm{s}}.}^{+},$ 5/2⁺, and 3/2⁺ states of ²⁵Ne are included for the β-decays.

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For the ²⁵Ne–²⁵Na, ²³F–²³Ne, and ²⁷Mg–²⁷Na pairs, general features of the neutrino energy loss and energy production are similar to the cases of ²⁵Na–²⁵Mg and ²³Ne–²³Na pairs. Averaged energy produced is suppressed by neutrino emissions above (below) the Urca densities for electron-capture (β-decay) processes.

When the production yield of ²⁴Mg from ¹²C + ¹²C reactions is small and the contribution of electron-capture reactions on ²⁴Mg to the evolution of stars is suppressed, the Urca processes for the present pairs can be important. In a recent model of O–Ne–Mg white dwarfs, these secondary coolings induced by the pairs give important effects on the evolution of the stars (Jones 2013; Denissenkov et al. 2015).

We briefly comment on electron-capture and β-decay rates for nuclear pairs with A = 20 and 24, ²⁰O–²⁰F, ²⁰F–²⁰Ne, ²⁴Ne–²⁴Na, and ²⁴Na–²⁴Mg. These weak rates are important for late stages of the evolution of stars with 8–10 M_⊙. Electron-capture reactions on ²⁴Mg and ²⁰Ne are important processes toward electron-capture supernova explosions or Fe core-collapse supernova explosions after contraction of the electron-degenerate O–Ne–Mg cores.

The weak rates for these nuclei are examined in detail in Martinez-Pinedo et al. (2014) by taking into account the forbidden transitions between ²⁰Ne (${0}_{{\rm{g}}.{\rm{s}}.}^{+}$) and ²⁰F (${2}_{{\rm{g}}.{\rm{s}}.}^{+}$). The transitions are found to give nonnegligible contributions at log₁₀ T < 9.0 in a density region: 9.3 < log₁₀(ρY_e) < 9.6. Calculated rates with the inclusion of the forbidden transitions are also tabulated in Suzuki et al. (2015). The rates were also examined in Takahara et al. (1989) with the use of USD and available experimental data, but the effects of the forbidden transitions were not taken into account. The GT strength for ${0}_{{\rm{g}}.{\rm{s}}.}^{+}\to$ 1⁺ (1.057 MeV) obtained from recent experimental (p, n) data on ²⁰Ne and used in Martinez-Pinedo et al. (2014) were also not taken into account as the data were not available in 1989. The GT strength, enhanced in the (p, n) data compared with the USD value, enhances the electron-capture rates on ²⁰Ne at log₁₀(ρY_e) > 9.6 by about 60%–70%.

As the Q-value for the ²⁴Ne–²⁴Na pair is as small as Q = 2.47 MeV owing to the pairing effects, electron-capture reactions on ²⁴Na occur sucessively after the electron-capture process on ²⁴Mg and contribute to heating the stars at later stages of evolutions, leading to the electron-capture supernova or Fe-core formation. The situation is the same as in the successive elecron-capture reaction on ²⁰F after the capture reaction on ²⁰Ne.

We have studied electron-capture and β-decay processes relevant to stars with O–Ne–Mg cores. Nuclear weak rates for sd-shell nuclei in stellar environments are evaluated by shell-model calculations with the use of the USDB Hamiltonian. The weak rates for nuclear pairs with A = 23 and 25, ²³Ne–²³Na and ²⁵Na–²⁵Mg, are important for nuclear Urca processes and cooling of the O–Ne–Mg core of the stars. Evaluations of the rates with fine meshes of density and temperature are required to get clear Urca densities (Toki et al. 2013). Coulomb effects due to electron screening and change of chemical potential of ions with electron background are evaluated for the weak transitions. The effects lead to enhancements of the threshold energy for electron-capture reactions and Q-value for the β-decays. The Coulomb corrections shift the Urca densities toward a higher-density region.

Triggering of electron-capture reactions is determined by the threshold energy of the transitions and chemical potential of electrons at high densities and temperatures. The capture reactions occur in order from pairs with smaller Q-values. After the first cooling of stars by the Urca processes for the pairs ²⁵Na–²⁵Mg and ²³Ne–²³Na, electron-capture reactions on ²⁴Mg, ²⁵Na, ²³Ne, and ²⁷Mg occur successively in this order as the density increases and the chemical potential of electron is increased and reach the corresponding threshold energy. Secondary cooling of stars by the Urca processes in these nuclei can be important at higher densities (Jones 2013).

The neutrino-energy-loss rate and γ-ray heating rate are also evaluated by shell-model calculations. The averaged neutrino energies and averaged energy production in the electron-capture and β-decay processes are obtained for the pairs, and the effects of energy loss by neutrino emissions are examined. Energy production is found to be suppressed by neutrino emissions above (below) the Urca densities for electron-capture (β-decay) processes.

Usually, the heating processes by electron-capture reactions on ²⁴Mg and ²⁰Ne occur after the cooling by the Urca processes in the ²⁵Na–²⁵Mg and ²³Ne–²³Na pairs. If this heating was not enough for some reason, for example, by a lack of ²⁴Mg, the secondary cooling by Urca processes in ²⁵Ne–²⁵Na, ²³F–²³Ne, and ²⁷Na–²⁷Mg pairs might have important roles in the evolution of stars (Jones 2013; Denissenkov et al. 2015).

Accurate evaluations of the nuclear weak rates at stellar environments are quite important for studies of the evolution of stars with 8–10 M_⊙. The weak rates determine the final fates of O–Ne–Mg cores and related white dwarfs but also a wider range of lower-mass stars with C–O cores, which contain primordial species of A = 20–27. For future advances for the studies, the electron-capture and β-decay rates, neutrino-energy-loss rates, and γ-ray heating rates for sd-shell nuclei with A = 17–28 have been evaluated with the Coulomb effects and tabulated for densities log₁₀(ρY_e) = 8.0–11.0 and temperatures log₁₀ T = 7.0–9.65 with fine meshes (Suzuki et al. 2015). The weak-rate data for the pairs with A = 23 and 25 have been applied to study the cooling of O–Ne–Mg cores by the nuclear Urca processes and evolution of stars with 8–10 M_⊙ (Jones et al. 2013). Stars with M_I = 8.8 M_⊙ are found to end up with electron-capture supernovae owing to the nuclear Urca processes, while stars with M_I = 9.5 M_⊙ are found to evolve to Fe core-collapse supernovae. Evolution of hybrid C–O–Ne white dwarfs as progenitors of Type Ia supernovae has also been studied with the nuclear Urca processes (Denissenkov et al. 2015). The hybrid progenitor models might explain the observed diversity of Type Ia supernovae.

We note from Figure 1 that the β-decay Q-value for ³¹Si $\to$ ³¹P is as small as 1.49 MeV. The β-decay Q-value for ²⁹Al $\to$ ²⁹Si is as small as 3.69 MeV. Electron-capture processes on ³¹P and ²⁹Si could have some important roles in the cooling and neutronization of the core in later O-burning and Si-burning stages in more massive stars.

The authors would like to thank S. Jones for useful communications on evolutions of stars with an O–Ne–Mg core. This work has been supported in part by Grants-in-Aid for Scientific Research (S)23224008, (S)21540267, (C)22540290, (C) 26400222, and (C) 15k05090, and the World Premier International Research Center Initiative of the MEXT of Japan.